Bin Yao Feifei Li Piyush Kumar Presenter Lian

Bin Yao, Feifei Li, Piyush Kumar Presenter: Lian Liu Reverse Furthest Neighbors in Spatial Databases

OUTLINE Introduction Related Work Algorithms - PFC (Progressive Furthest Cell) - CHFC (Convex Hull Furthest Cell) Experiment Discussion

INTRODUCTION Assume you live at p 1 (p 2, p 3), where would you prefer to build a chemical factory among q 1~q 3?

INTRODUCTION

INTRODUCTION Let P={p 1, p 2, p 3} Q={q 1, q 2, q 3} fn(p 1, Q)=q 3 fn(p 2, Q)=q 1 fn(p 3, Q)=q 1 BRFN(q 1, Q, P)={p 2, p 3} BRFN(q 2, Q, P)={} BRFN(q 3, Q, P)={p 1} Build the chemical factory here

INTRODUCTION Problem: Given query point q, data set P (and Q), Compute MRFN(q, P) and BRFN(q, Q, P). MRFN Furthest neighbor Voronoi cell & Voronoi diagram MRFN BRFN PFC CHFC for BRFN CHFC

RELATED WORK MBR (Minimum Bounding Rectangles) has 3 important distances to a point: Min Distance Max Distance Minmax Distance

RELATED WORK R-tree is an index data structure. In R-trees, points are grouped into MBRs, which are recursively grouped into MBRs in higher levels of the tree.

RELATED WORK Range query: retrieves all points that locates within the query window. R-tree based algorithms proves to be efficient to deal with range queries.

MRFN How to compute the MRFN of a given query point? BFS (Brute-Force Search) PFC (Progressive Furthest Cell) Main Idea: How to compute? 1. Find the cell (region) in which all reverse furthest neighbors of the query point located 2. Perform a range query with the cell

PFC FVC (Furthest Voronoi Cell)

PFC FVC Example: query point = q 1 fvc(q 1, P)

PFC (Progressive Furthest Cell) Algorithm Points and MBRs are stored in a priority queue L with their minmaxdist sorted in decreasing order. Two vectors Vc and Vp are also maintained: Vc: Furthest neighbor candidates Vp: Disqualifying points

PFC – mechanism fvc(q)={} e is a point pop e out from L until L is empty e is an MBR update fvc(q) e∈fvc(q) e∩fvc(q)={} compute e∩fvc(q)≠{} return {} push e into Vc push e into Vp examine each child c At last, we update fvc(q) using Vp and then filter points in Vc using fvc(q) c∩fvc(q)≠{} push c into L push c into Vp c∩fvc(q)={}

PFC Example: fvc(q) L={p 1, R 1} Vc={} Vp={} L={R 1} Vc={p 1} Vp={}

PFC Example: fvc(q) L={p 3} Vc={p 1} Vp={p 2} fvc(q) L={} Vc={p 1, p 3} Vp={p 2}

PFC Example: Finally, we use all points in Vp (i. e. p 2) to update fvc(q). Then, we perform a range query using the updated fvc(q). The result is {p 3}。 fvc(q) MRFN(q)={p 3}

PFC Efficiency of PFC makes fvc(q) quickly shrink. If the query point does not have any reverse furthest neighbors, Φ will quickly be reported. However, it is still not efficient enough. Improvement: CHFC algorithm.

CHFC Convex Hull The Convex Hull of a set of points P is the smallest convex polygon that fully contains P. Denoted as CP.

CHFC Lemma: Given a point set P and its convex hull Cp, for a point q, let p*=fn(q, P), then p*∈CP. fvc(p, P)=fvc(p, CP)

CHFC (Convex Hull Furthest Cell) Given a set of points P and a query point p: Compute CP∪{p} Compute fvc(p, P) using CP∪{p} Perform a range query with fvc(p, P)

CHFC BRFN (Bichromatic Reverse Furthest Neighbor) can be found in the same way as MRFN. The only one difference is, we compute fvc(q, Q, P) will Q, can perform range query in P.

CHFC Efficiency of CHFC: For most (but not all) cases, |CP| << |P|. That is, the number of points considered are likely to be greatly reduced. Difficulty: How to compute and update CP when |P| is very large and even |CP| cannot fit into memory.

CHFC Computing Convex Hull Convex hulls can be found in either a distance -first or a depth-first manner. Distance-first approach is optimal in the number of page accesses, and the complexity is O(nlogn). Depth-first algorithms can run in O(n) time for worst case, but not optimal in disk accessing.

CHFC Updating Convex Hull Inserting new points: Lemma: P is a point set. If point q is contained by CP , CP∪{q} =CP Otherwise, CP∪{q} =CCp∪{q}

CHFC Updating Convex Hull Deleting points: Points or MBRs with the largest perpendicular distance to plpr are added into CP first, until there is no points outside the current convex hull.

CHFC External Convex Hull Computing Existing algorithms can found 2 -Dimensional convex hulls with I/Os. However, when convex hulls are still too large to fit into memory, we use Dudley’s approximate convex hull.

EXPERIMENT CPU time & number of IOs

DISCUSSION Thank You! Questions?